TITLE: What are conditions on real coefficients for zeros of a polynomial to be on the unit circle?
QUESTION [11 upvotes]: My complex analysis is decades in the rear view mirror.  Perhaps someone here can help.  I am looking for necessary and sufficient conditions on the coefficients of of a real polynomial of one complex variable (i.e. an element of ℝ[z]) so that all of its zeros will be on the unit circle.  Clearly it is necessary that the polynomial is palindromic, but that is not sufficient.  My question arose, after I discovered that if r,s are both real numbers then the polynomial p(z) = 

has all its roots on the unit circle.  Since I had not seen anything like this before, I wondered if there were perhaps just some conditions on the coefficients one could check.

REPLY [4 votes]: See paper
Palindrome-Polynomials with Roots on the Unit Circle by John Konvalina and Valentin Matache